lecture07 - MA1100 Lecture 7 Mathematical Proofs Parity of...

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1 MA1100 Lecture 7 Mathematical Proofs Parity of integers Divisibility of integers Direct Proofs Proof by Contrapositive Chartrand: section 3.2, 3.3, 4.1
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Lecture 7 2 True Statements • Mathematical statements that are true are known as mathematical facts . •A proposition is a true mathematical statement that has a proof . • A proposition that is “ important ” is called a theorem . • A proposition that is used in the proof of a theorem is called a lemma . • A proposition that follows (easily) from a theorem is called a corollary . • There are true statements that do not have proofs e.g. definitions , axioms . Chartrand calls it “Result”
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Lecture 7 3 An Example If x is an even integer, then x 2 is divisible by 4. Proposition We want to give a proof for this (universal) conditional statement. We need to : know the definitions of even integers and divisibility use some basic facts about integers use algebraic manipulation
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Lecture 7 4 Definition Examples (Mathematical) 1. Even and odd integers 2.n is divisible by m A (mathematical) definition gives the precise meaning of a word or phrase that represents some object, property or other concepts.
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Lecture 7 5 Even integers 2 6 -14 Example = 2 ä 1 = 2 ä 3 = 2 ä (-7) An integer a is an even integer if there exists an integer n such that a = 2n . Definition In general, an even integer can be written in the form 2n where n is an integer
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Lecture 7 6 Odd integers 3 7 -13 Example = 2 ä 1 + 1 = 2 ä 3 + 1 = 2 ä (-7) + 1 An integer a is an odd integer if there exists an integer n such that a = 2n + 1 . Definition In general, an odd integer can be written in the form 2n + 1 where n is an integer
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Lecture 7 7 Even and odd integers Why do we need definitions for even and odd integers? Is 0 an even integer? Not to identify specific even and odd integers To describe general even and odd integers To be used in the proof of statements involving even and odd integers
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Lecture 7 8 Undefined Terms Some basic concepts remain undefined . Numbers, integers, … Addition, multiplication, … Point, line, … Examples
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Lecture 7 9 Divisibility Definition Let m and n be integers. We say m divides n if there exists an integer q such that n = mq . Example 3 divides 12 as 12 = 3 ä 4 7 divides 21 as 21 = 7 ä 3 In general, when m divides n, n can be written in the form m ä k where k is an integer
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Lecture 7 10 Divisibility Alternative names We also say: ± m is a divisor of n ± m is a factor of n ± n is a
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This note was uploaded on 04/15/2010 for the course MATHS MA1101R taught by Professor Vt during the Spring '10 term at National University of Singapore.

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lecture07 - MA1100 Lecture 7 Mathematical Proofs Parity of...

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